Snide is so close! (And much better than anything I found.) It's still a bit too strong. People who choose to do this are masters at it; they make it so subtle that even if someone overhears it, they will probably not tell them they are being rude. It's like a "subtle passive-aggressive rudeness".
I'm going to hazard arch, which is used like @Jon's applicable *patronizing@Jon's applicable *patronizing*, but with a subtler tone of snarkiness.
From Wordnet:
arch (adjective): (used of behavior or attitude) characteristic of those who treat others with condescension
And from Collins:
arch: knowing or superior
An arch tone is not quite a cliche, but very common, and used when the social situation is recognized by both parties as licensing the superior attitude, such was when the person using it is from a higher social class (back in the days when those were more explicit and subscribed to), or today when the other party has screwed up in a visible manner.
In short, think old-school British upper class dismissively ordering about or correcting someone they think beneath themselves.
Alternatives include superior (not as snarky or witty) and condescending (more explicit and often more brutal, and sometimes felt by the other party to be unjustified).